?
G = C14.1212+ (1+4) order 448 = 26·7
30th non-split extension by C14 of 2+ (1+4) acting via 2+ (1+4)/C4○D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.1212+ (1+4), C7⋊D4⋊6D4, C4⋊C4⋊15D14, C7⋊8(D4⋊5D4), C28⋊1D4⋊30C2, C22⋊C4⋊15D14, D14.22(C2×D4), (C22×C4)⋊23D14, D14⋊10(C4○D4), D14⋊D4⋊29C2, D28⋊C4⋊31C2, C22⋊D28⋊19C2, C22.12(D4×D7), D14⋊C4⋊22C22, D14⋊Q8⋊27C2, (C2×D4).163D14, (C2×D28)⋊26C22, (C2×C28).71C23, Dic7.26(C2×D4), C14.83(C22×D4), C22.D4⋊4D7, Dic7⋊D4⋊20C2, Dic7⋊4D4⋊18C2, D14.5D4⋊27C2, (C2×C14).198C24, Dic7⋊C4⋊22C22, (C22×C28)⋊17C22, (C4×Dic7)⋊31C22, C2.41(D4⋊8D14), C23.D7⋊30C22, C23.26(C22×D7), Dic7.D4⋊31C2, (C2×Dic14)⋊55C22, (D4×C14).136C22, (C22×C14).33C23, (C23×D7).57C22, C22.219(C23×D7), (C2×Dic7).102C23, (C22×Dic7)⋊24C22, (C22×D7).206C23, (C2×D4×D7)⋊15C2, C2.56(C2×D4×D7), C2.60(D7×C4○D4), (C2×C4×D7)⋊21C22, (C2×D14⋊C4)⋊23C2, (C2×C4○D28)⋊11C2, (C7×C4⋊C4)⋊25C22, (D7×C22⋊C4)⋊12C2, (C2×C14).59(C2×D4), C14.172(C2×C4○D4), (C2×C7⋊D4)⋊41C22, (C2×C4).61(C22×D7), (C7×C22⋊C4)⋊21C22, (C7×C22.D4)⋊6C2, SmallGroup(448,1107)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1932 in 334 conjugacy classes, 105 normal (91 characteristic)
C1, C2 [×3], C2 [×9], C4 [×10], C22, C22 [×2], C22 [×27], C7, C2×C4 [×5], C2×C4 [×14], D4 [×18], Q8 [×2], C23 [×2], C23 [×14], D7 [×6], C14 [×3], C14 [×3], C42, C22⋊C4 [×3], C22⋊C4 [×9], C4⋊C4 [×2], C4⋊C4 [×2], C22×C4, C22×C4 [×5], C2×D4, C2×D4 [×12], C2×Q8, C4○D4 [×4], C24 [×2], Dic7 [×2], Dic7 [×3], C28 [×5], D14 [×4], D14 [×18], C2×C14, C2×C14 [×2], C2×C14 [×5], C2×C22⋊C4 [×2], C4×D4 [×2], C22≀C2 [×2], C4⋊D4 [×3], C22⋊Q8, C22.D4, C22.D4, C4.4D4, C22×D4, C2×C4○D4, Dic14 [×2], C4×D7 [×7], D28 [×7], C2×Dic7 [×4], C2×Dic7, C7⋊D4 [×4], C7⋊D4 [×5], C2×C28 [×5], C2×C28 [×2], C7×D4 [×2], C22×D7 [×4], C22×D7 [×10], C22×C14 [×2], D4⋊5D4, C4×Dic7, Dic7⋊C4 [×2], D14⋊C4 [×8], C23.D7, C7×C22⋊C4 [×3], C7×C4⋊C4 [×2], C2×Dic14, C2×C4×D7 [×4], C2×D28 [×4], C4○D28 [×4], D4×D7 [×4], C22×Dic7, C2×C7⋊D4 [×4], C22×C28, D4×C14, C23×D7 [×2], D7×C22⋊C4, Dic7⋊4D4, C22⋊D28 [×2], D14⋊D4, Dic7.D4, D28⋊C4, D14.5D4, C28⋊1D4, D14⋊Q8, C2×D14⋊C4, Dic7⋊D4, C7×C22.D4, C2×C4○D28, C2×D4×D7, C14.1212+ (1+4)
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C4○D4 [×2], C24, D14 [×7], C22×D4, C2×C4○D4, 2+ (1+4), C22×D7 [×7], D4⋊5D4, D4×D7 [×2], C23×D7, C2×D4×D7, D7×C4○D4, D4⋊8D14, C14.1212+ (1+4)
Generators and relations
G = < a,b,c,d,e | a14=b4=e2=1, c2=a7, d2=b2, ab=ba, cac-1=dad-1=a-1, ae=ea, cbc-1=a7b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d >
►On 112 points
(1 2 3 4 5 6 7 8 9 10 11 12 13 14)(15 16 17 18 19 20 21 22 23 24 25 26 27 28)(29 30 31 32 33 34 35 36 37 38 39 40 41 42)(43 44 45 46 47 48 49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64 65 66 67 68 69 70)(71 72 73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96 97 98)(99 100 101 102 103 104 105 106 107 108 109 110 111 112)
(1 50 85 62)(2 51 86 63)(3 52 87 64)(4 53 88 65)(5 54 89 66)(6 55 90 67)(7 56 91 68)(8 43 92 69)(9 44 93 70)(10 45 94 57)(11 46 95 58)(12 47 96 59)(13 48 97 60)(14 49 98 61)(15 40 73 103)(16 41 74 104)(17 42 75 105)(18 29 76 106)(19 30 77 107)(20 31 78 108)(21 32 79 109)(22 33 80 110)(23 34 81 111)(24 35 82 112)(25 36 83 99)(26 37 84 100)(27 38 71 101)(28 39 72 102)
(1 43 8 50)(2 56 9 49)(3 55 10 48)(4 54 11 47)(5 53 12 46)(6 52 13 45)(7 51 14 44)(15 37 22 30)(16 36 23 29)(17 35 24 42)(18 34 25 41)(19 33 26 40)(20 32 27 39)(21 31 28 38)(57 90 64 97)(58 89 65 96)(59 88 66 95)(60 87 67 94)(61 86 68 93)(62 85 69 92)(63 98 70 91)(71 102 78 109)(72 101 79 108)(73 100 80 107)(74 99 81 106)(75 112 82 105)(76 111 83 104)(77 110 84 103)
(1 24 85 82)(2 23 86 81)(3 22 87 80)(4 21 88 79)(5 20 89 78)(6 19 90 77)(7 18 91 76)(8 17 92 75)(9 16 93 74)(10 15 94 73)(11 28 95 72)(12 27 96 71)(13 26 97 84)(14 25 98 83)(29 68 106 56)(30 67 107 55)(31 66 108 54)(32 65 109 53)(33 64 110 52)(34 63 111 51)(35 62 112 50)(36 61 99 49)(37 60 100 48)(38 59 101 47)(39 58 102 46)(40 57 103 45)(41 70 104 44)(42 69 105 43)
(1 75)(2 76)(3 77)(4 78)(5 79)(6 80)(7 81)(8 82)(9 83)(10 84)(11 71)(12 72)(13 73)(14 74)(15 97)(16 98)(17 85)(18 86)(19 87)(20 88)(21 89)(22 90)(23 91)(24 92)(25 93)(26 94)(27 95)(28 96)(29 63)(30 64)(31 65)(32 66)(33 67)(34 68)(35 69)(36 70)(37 57)(38 58)(39 59)(40 60)(41 61)(42 62)(43 112)(44 99)(45 100)(46 101)(47 102)(48 103)(49 104)(50 105)(51 106)(52 107)(53 108)(54 109)(55 110)(56 111)
G:=sub<Sym(112)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14)(15,16,17,18,19,20,21,22,23,24,25,26,27,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42)(43,44,45,46,47,48,49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98)(99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,50,85,62)(2,51,86,63)(3,52,87,64)(4,53,88,65)(5,54,89,66)(6,55,90,67)(7,56,91,68)(8,43,92,69)(9,44,93,70)(10,45,94,57)(11,46,95,58)(12,47,96,59)(13,48,97,60)(14,49,98,61)(15,40,73,103)(16,41,74,104)(17,42,75,105)(18,29,76,106)(19,30,77,107)(20,31,78,108)(21,32,79,109)(22,33,80,110)(23,34,81,111)(24,35,82,112)(25,36,83,99)(26,37,84,100)(27,38,71,101)(28,39,72,102), (1,43,8,50)(2,56,9,49)(3,55,10,48)(4,54,11,47)(5,53,12,46)(6,52,13,45)(7,51,14,44)(15,37,22,30)(16,36,23,29)(17,35,24,42)(18,34,25,41)(19,33,26,40)(20,32,27,39)(21,31,28,38)(57,90,64,97)(58,89,65,96)(59,88,66,95)(60,87,67,94)(61,86,68,93)(62,85,69,92)(63,98,70,91)(71,102,78,109)(72,101,79,108)(73,100,80,107)(74,99,81,106)(75,112,82,105)(76,111,83,104)(77,110,84,103), (1,24,85,82)(2,23,86,81)(3,22,87,80)(4,21,88,79)(5,20,89,78)(6,19,90,77)(7,18,91,76)(8,17,92,75)(9,16,93,74)(10,15,94,73)(11,28,95,72)(12,27,96,71)(13,26,97,84)(14,25,98,83)(29,68,106,56)(30,67,107,55)(31,66,108,54)(32,65,109,53)(33,64,110,52)(34,63,111,51)(35,62,112,50)(36,61,99,49)(37,60,100,48)(38,59,101,47)(39,58,102,46)(40,57,103,45)(41,70,104,44)(42,69,105,43), (1,75)(2,76)(3,77)(4,78)(5,79)(6,80)(7,81)(8,82)(9,83)(10,84)(11,71)(12,72)(13,73)(14,74)(15,97)(16,98)(17,85)(18,86)(19,87)(20,88)(21,89)(22,90)(23,91)(24,92)(25,93)(26,94)(27,95)(28,96)(29,63)(30,64)(31,65)(32,66)(33,67)(34,68)(35,69)(36,70)(37,57)(38,58)(39,59)(40,60)(41,61)(42,62)(43,112)(44,99)(45,100)(46,101)(47,102)(48,103)(49,104)(50,105)(51,106)(52,107)(53,108)(54,109)(55,110)(56,111)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14)(15,16,17,18,19,20,21,22,23,24,25,26,27,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42)(43,44,45,46,47,48,49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98)(99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,50,85,62)(2,51,86,63)(3,52,87,64)(4,53,88,65)(5,54,89,66)(6,55,90,67)(7,56,91,68)(8,43,92,69)(9,44,93,70)(10,45,94,57)(11,46,95,58)(12,47,96,59)(13,48,97,60)(14,49,98,61)(15,40,73,103)(16,41,74,104)(17,42,75,105)(18,29,76,106)(19,30,77,107)(20,31,78,108)(21,32,79,109)(22,33,80,110)(23,34,81,111)(24,35,82,112)(25,36,83,99)(26,37,84,100)(27,38,71,101)(28,39,72,102), (1,43,8,50)(2,56,9,49)(3,55,10,48)(4,54,11,47)(5,53,12,46)(6,52,13,45)(7,51,14,44)(15,37,22,30)(16,36,23,29)(17,35,24,42)(18,34,25,41)(19,33,26,40)(20,32,27,39)(21,31,28,38)(57,90,64,97)(58,89,65,96)(59,88,66,95)(60,87,67,94)(61,86,68,93)(62,85,69,92)(63,98,70,91)(71,102,78,109)(72,101,79,108)(73,100,80,107)(74,99,81,106)(75,112,82,105)(76,111,83,104)(77,110,84,103), (1,24,85,82)(2,23,86,81)(3,22,87,80)(4,21,88,79)(5,20,89,78)(6,19,90,77)(7,18,91,76)(8,17,92,75)(9,16,93,74)(10,15,94,73)(11,28,95,72)(12,27,96,71)(13,26,97,84)(14,25,98,83)(29,68,106,56)(30,67,107,55)(31,66,108,54)(32,65,109,53)(33,64,110,52)(34,63,111,51)(35,62,112,50)(36,61,99,49)(37,60,100,48)(38,59,101,47)(39,58,102,46)(40,57,103,45)(41,70,104,44)(42,69,105,43), (1,75)(2,76)(3,77)(4,78)(5,79)(6,80)(7,81)(8,82)(9,83)(10,84)(11,71)(12,72)(13,73)(14,74)(15,97)(16,98)(17,85)(18,86)(19,87)(20,88)(21,89)(22,90)(23,91)(24,92)(25,93)(26,94)(27,95)(28,96)(29,63)(30,64)(31,65)(32,66)(33,67)(34,68)(35,69)(36,70)(37,57)(38,58)(39,59)(40,60)(41,61)(42,62)(43,112)(44,99)(45,100)(46,101)(47,102)(48,103)(49,104)(50,105)(51,106)(52,107)(53,108)(54,109)(55,110)(56,111) );
G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14),(15,16,17,18,19,20,21,22,23,24,25,26,27,28),(29,30,31,32,33,34,35,36,37,38,39,40,41,42),(43,44,45,46,47,48,49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64,65,66,67,68,69,70),(71,72,73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96,97,98),(99,100,101,102,103,104,105,106,107,108,109,110,111,112)], [(1,50,85,62),(2,51,86,63),(3,52,87,64),(4,53,88,65),(5,54,89,66),(6,55,90,67),(7,56,91,68),(8,43,92,69),(9,44,93,70),(10,45,94,57),(11,46,95,58),(12,47,96,59),(13,48,97,60),(14,49,98,61),(15,40,73,103),(16,41,74,104),(17,42,75,105),(18,29,76,106),(19,30,77,107),(20,31,78,108),(21,32,79,109),(22,33,80,110),(23,34,81,111),(24,35,82,112),(25,36,83,99),(26,37,84,100),(27,38,71,101),(28,39,72,102)], [(1,43,8,50),(2,56,9,49),(3,55,10,48),(4,54,11,47),(5,53,12,46),(6,52,13,45),(7,51,14,44),(15,37,22,30),(16,36,23,29),(17,35,24,42),(18,34,25,41),(19,33,26,40),(20,32,27,39),(21,31,28,38),(57,90,64,97),(58,89,65,96),(59,88,66,95),(60,87,67,94),(61,86,68,93),(62,85,69,92),(63,98,70,91),(71,102,78,109),(72,101,79,108),(73,100,80,107),(74,99,81,106),(75,112,82,105),(76,111,83,104),(77,110,84,103)], [(1,24,85,82),(2,23,86,81),(3,22,87,80),(4,21,88,79),(5,20,89,78),(6,19,90,77),(7,18,91,76),(8,17,92,75),(9,16,93,74),(10,15,94,73),(11,28,95,72),(12,27,96,71),(13,26,97,84),(14,25,98,83),(29,68,106,56),(30,67,107,55),(31,66,108,54),(32,65,109,53),(33,64,110,52),(34,63,111,51),(35,62,112,50),(36,61,99,49),(37,60,100,48),(38,59,101,47),(39,58,102,46),(40,57,103,45),(41,70,104,44),(42,69,105,43)], [(1,75),(2,76),(3,77),(4,78),(5,79),(6,80),(7,81),(8,82),(9,83),(10,84),(11,71),(12,72),(13,73),(14,74),(15,97),(16,98),(17,85),(18,86),(19,87),(20,88),(21,89),(22,90),(23,91),(24,92),(25,93),(26,94),(27,95),(28,96),(29,63),(30,64),(31,65),(32,66),(33,67),(34,68),(35,69),(36,70),(37,57),(38,58),(39,59),(40,60),(41,61),(42,62),(43,112),(44,99),(45,100),(46,101),(47,102),(48,103),(49,104),(50,105),(51,106),(52,107),(53,108),(54,109),(55,110),(56,111)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 10 | 0 | 0 |
| 0 | 0 | 9 | 23 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 5 | 7 | 0 | 0 | 0 | 0 |
| 9 | 24 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 5 | 7 | 0 | 0 | 0 | 0 |
| 17 | 24 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 22 | 0 | 0 |
| 0 | 0 | 13 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 0 | 0 | 17 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 22 | 0 | 0 |
| 0 | 0 | 13 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 2 |
| 0 | 0 | 0 | 0 | 28 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 27 |
| 0 | 0 | 0 | 0 | 0 | 28 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,9,9,0,0,0,0,10,23,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[5,9,0,0,0,0,7,24,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[5,17,0,0,0,0,7,24,0,0,0,0,0,0,11,13,0,0,0,0,22,18,0,0,0,0,0,0,17,0,0,0,0,0,0,17],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,11,13,0,0,0,0,22,18,0,0,0,0,0,0,28,28,0,0,0,0,2,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,27,28] >;
67 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 14P | 14Q | 14R | 28A | ··· | 28L | 28M | ··· | 28U |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | 14 | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 14 | 14 | 14 | 14 | 28 | 28 | 2 | 2 | 4 | 4 | 4 | 4 | 14 | 14 | 14 | 14 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
67 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | C4○D4 | D14 | D14 | D14 | D14 | 2+ (1+4) | D4×D7 | D7×C4○D4 | D4⋊8D14 |
| kernel | C14.1212+ (1+4) | D7×C22⋊C4 | Dic7⋊4D4 | C22⋊D28 | D14⋊D4 | Dic7.D4 | D28⋊C4 | D14.5D4 | C28⋊1D4 | D14⋊Q8 | C2×D14⋊C4 | Dic7⋊D4 | C7×C22.D4 | C2×C4○D28 | C2×D4×D7 | C7⋊D4 | C22.D4 | D14 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C14 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 3 | 4 | 9 | 6 | 3 | 3 | 1 | 6 | 6 | 6 |
In GAP, Magma, Sage, TeX
C_{14}._{121}2_+^{(1+4)} % in TeX
G:=Group("C14.121ES+(2,2)"); // GroupNames label
G:=SmallGroup(448,1107);
// by ID
G=gap.SmallGroup(448,1107);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,758,100,346,297,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=e^2=1,c^2=a^7,d^2=b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,a*e=e*a,c*b*c^-1=a^7*b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=b^2*d>;
// generators/relations